Square-root cancellation for sums of coefficients of $\operatorname{GL}(m)$-automorphic forms over values of random polynomials

Technical Summary: Square-Root Cancellation for Sums of Coefficients of $GL_m$-Automorphic Forms Over Values of Random Polynomials

Problem Statement
The paper addresses the problem of estimating sums of Fourier coefficients of automorphic forms evaluated along polynomial sequences. Specifically, given an automorphic irreducible cuspidal representation $\pi$ of $GL_m$ over $\mathbb{Q}$ with unitary central character, and a polynomial $P \in \mathbb{Z}[t]$ of degree $d$, the authors investigate the sum:
$$S_P(x) = \sum_{\substack{n \le x \\ P(n) > 0}} \lambda_\pi(P(n))$$
where $\lambda_\pi(n)$ denotes the normalized Fourier coefficients of $\pi$. The central question is whether these sums exhibit "square-root cancellation," i.e., whether $S_P(x) = O(x^{1/2})$, when averaged over a family of random polynomials.

While the case of linear polynomials ($d=1$) reduces to mean value estimates of Fourier coefficients (a classical problem addressed via $L$-functions and summation formulas), the general case for $d \ge 2$ is considered intractable for specific polynomials. Previous work has established bounds for specific quadratic polynomials and $GL_2$ forms (e.g., Blomer, Templier, Templier–Tsimerman), often relying on trace formulas or sup-norm bounds with error terms depending on the Selberg eigenvalue conjecture.

Methodology
The authors adopt a probabilistic approach, ordering polynomials by their height $|P|$ (the maximum absolute value of their coefficients) and averaging over the set of polynomials with degree $d$ and height $\le H$. The methodology proceeds through three main analytical stages:

1. Second Moment Analysis: The core of the proof involves estimating the second moment of the sum over the family of polynomials:
   $$\frac{1}{\#\mathcal{P}} \sum_{P \in \mathcal{P}} \left| \sum_{n \le x} \lambda_\pi(P(n)) \right|^2$$
   where $\mathcal{P} = \{P \in \mathbb{Z}[t] : \deg(P)=d, |P| \le H\}$.

2. Circle Method and Exponential Sums:

   • With Minor Arcs (Asymptotic Results): For $GL_2$ and $GL_3$, the authors utilize the Circle Method. They expand the square of the sum and separate diagonal ($n_1 = n_2$) and non-diagonal ($n_1 \neq n_2$) terms. The non-diagonal terms are handled using bounds on twisted linear sums of the form $\sum \lambda_\pi(n) e^{2\pi i \gamma n}$.
     • For $GL_2$, they employ Wilton's bound.
     • For $GL_3$, they utilize results by Miller on additive twists.
     • For $GL_m$ ($m \ge 2$), they use bounds by Jiang, Lü, and Wang.
   • Without Minor Arcs (Upper Bounds): For general $m$, the authors relax the assumptions to avoid the need for specific asymptotic bounds on twisted sums. They introduce Gauss kernels (Jacobi theta functions) to smooth the coefficients of the random polynomials. This allows them to apply the Circle Method under weaker conditions on the exponential sums, specifically requiring only a uniform bound of the form $\sup_{a/q} |\sum \lambda_\pi(n) e^{2\pi i (a/q) n}| \ll q^\sigma x^\tau$.

3. Rankin–Selberg Theory: The diagonal terms in the second moment are evaluated using the asymptotic behavior of the second moment of the coefficients themselves, $\sum_{n \le x} |\lambda_\pi(n)|^2 \sim C_\pi x$, derived from the Rankin–Selberg $L$-function $L(s, \pi \times \tilde{\pi})$.

Key Contributions and Results

1. General Square-Root Cancellation (Theorem 1.1):
   The authors establish unconditionally that for any $GL_m$ cusp form ($m \ge 2$) and for almost all polynomials of arbitrary degree $d$, the sum satisfies square-root cancellation. Specifically, for $x \le H^{\alpha - \varepsilon}$ (where $\alpha$ depends on $m$ and $d$), the average of the squared sums is bounded by $O(x)$. This implies that for a generic polynomial, the sum is $O(x^{1/2})$.

   • The exponent $\alpha$ is given by $\min \left( \frac{2}{2d(2m+3) + 3m + 4}, \frac{2}{d(m^2-1)} \right)$.

2. Asymptotic Formulas for $GL_2$ and $GL_3$ (Theorem 1.2):
   For $m=2$ and $m=3$, the authors prove a stronger result: an asymptotic formula for the second moment.
   $$\frac{1}{\#\mathcal{P}} \sum_{P \in \mathcal{P}} \left| \sum_{n \le x} \lambda_\pi(P(n)) \right|^2 = C_\pi x + O\left( \frac{x}{(\log x)^N} \right)$$
   This holds for $x \le H^{\beta(m) - \varepsilon}$, where $\beta(2) = \frac{1}{2+3d}$ and $\beta(3) = \frac{1}{7d+4}$. This confirms that the random sums behave like a random walk with variance proportional to $x$.

3. General Theorem on Arithmetic Functions (Theorem 3.1 & 4.1):
   The paper provides a general framework (Theorem 3.1) for the second moment of sums of arbitrary arithmetic functions $g(n)$ over random polynomials, provided $g$ satisfies specific bounds on its twisted sums and its second moment. This is noted as the first result in the literature giving asymptotics for the second moment of general arithmetic functions over values of random polynomials.

Significance and Claims
The paper claims to provide the first unconditional proof of square-root cancellation for sums of $GL_m$-automorphic coefficients over values of random polynomials of arbitrary degree. By shifting the focus from specific polynomials to a "generic" setting, the authors bypass the intractability of the general problem.

The authors highlight that while recent progress on specific $GL_2$ cases (by Huang, Dasgupta–Leung–Young) improves error terms for fixed polynomials, these do not immediately improve the exponent $\beta(2)$ in their random polynomial setting. The significance lies in the unconditionality of the result (no reliance on the Selberg eigenvalue conjecture) and the extension to higher rank groups ($GL_m$ for $m \ge 2$).

The work distinguishes itself from related problems involving absolute values of coefficients (which typically only allow logarithmic savings) by focusing on the signed sums, where cancellation is expected. The results rely on feeding known bounds for twisted linear sums (from Jiang–Lü–Wang, Miller, Wilton) into a novel probabilistic framework involving the Circle Method and Gauss kernels.